The number of votes, votes, can be smaller than the number of ballots, ballots, because a vote is not recorded if (1) the person fails to vote for President, (2) votes for more than one candidate, or (3) the equipment fails to record the vote.
We are interested in the undercount, which is defined as (ballots - votes) / ballots. Does it depend on the type of voting machine equip, economy econ, percentage of African Americans perAA, whether the county is rural or urban rural, or whether the county is part of Atlanta metropolitan area atlanta.
Let’s further stratify according to factors equip, econ, usage, and atlanta.
for (var inc("equip", "econ", "usage", "atlanta")) { plot <-ggplot(data = gavote) +geom_point(mapping =aes(x = perAA, y = undercount, color =get(var)))print(plot)}
For qualitative predictors, we can also do side-by-side box plots to reveal potential relationship with responses.
What does boxplot show us?
Any other plots do you usually use beside boxplot?
Violin plot
for (var inc("equip", "econ", "usage", "atlanta")) { plot <-ggplot(data = gavote) +geom_boxplot(mapping =aes(x =get(var), y = undercount)) +xlab(var)print(plot)}
Pairwise scatter plots are useful to explore the relationships between several continuous variables. We use the ggpairs function from GGally package (GGally examples).
To formally study how undercount is affected by other variables, we postulate a linear model \[
Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_{p-1} X_{p-1} + \epsilon,
\] where
the response variable or dependent variable\(Y\) is the undercount, the variable of interest,
the predictors\(X_1,\ldots,X_{p-1}\) encodes the predictors perAA, econ, etc.,
the regression coefficients or parameters\(\beta_0, \ldots, \beta_{p-1}\) reflects the effects of predictors on response variable \(Y\), especially \(\beta_0\) is called the intercept, and
the error term\(\epsilon\) includes measurement error.
The least squares estimate of \(\boldsymbol{\beta}\), called \(\widehat{\boldsymbol{\beta}}\), minimizes \[
\|\boldsymbol{\epsilon}\|^2 = \sum_i \epsilon_i^2 = \boldsymbol{\epsilon}^T \boldsymbol{\epsilon} = (\mathbf{y} - \mathbf{X} \boldsymbol{\beta})^T (\mathbf{y} - \mathbf{X} \boldsymbol{\beta}) = \|\mathbf{y} - \mathbf{X} \boldsymbol{\beta}\|^2.
\]
It’s important to keep in mind that the least squares estimate only assumes that
\(\mathbb{E}(\mathbf{Y}) = \mathbf{X} \boldsymbol{\beta}\) (errors have mean 0), and
\(\text{Var}(\mathbf{Y}) = \sigma_0^2 \mathbf{I}_n\) (errors are independent with common variance).
Under these assumptions, the celebrated Gauss-Markov theorem states that the least squares estimate is the best (smallest variance) linear unbiased estimates.
It is the inference part (p-value, standard errors, confidence intervals) that uses the normality assumption
Under normality assumption, the least squares estimate coincides with the maximum likelihood estimate (MLE).
4 A model with two quantitative predictors
We start with a linear model with just two predictors: percentage of Gore votes, pergore, and percentage of African Americans, perAA. \[
\text{undercount} = \beta_0 + \beta_1 \cdot \text{pergore} + \beta_2 \cdot \text{perAA} + \epsilon.
\]
An unbiased estimate of the error variance \(\sigma^2\) is \[
\widehat{\sigma} = \sqrt{\frac{\text{RSS}}{\text{df}}}
\]
sqrt(deviance(lmod) /df.residual(lmod))
[1] 0.02444895
lmodsum$sigma
[1] 0.02444895
A commonly used goodness of fit measure is \(R^2\), or coefficient of determination or percentage of variance explained\[
R^2 = 1 - \frac{\sum_i (y_i - \widehat{y}_i)^2}{\sum_i (y_i - \bar{y})^2} = 1 - \frac{\text{RSS}}{\text{TSS}},
\] where \(\text{TSS} = \sum_i (y_i - \bar{y})^2\) is the total sum of squares.
lmodsum$r.squared
[1] 0.05308861
An \(R^2\) of about 5% indicates the model has a poor fit. \(R^2\) can also be interpreted as the (squared) correlation between the predicted values and the response
cor(predict(lmod), gavote$undercount)^2
[1] 0.05308861
Add more predictors into a model always increase \(R^2\). The adjusted \(R^2\) adjusts to the fact that a larger model also uses more parameters. Adding a predictor will only increase \(R_a^2\) if it has some predictive value. \[
R_a^2 = 1 - \frac{\text{RSS} / (n - p)}{\text{TSS} / (n - 1)}.
\]
lmodsum$adj.r.squared
[1] 0.04094872
5 A model with both quantitative and qualitative predictors
Now we also want to include factors equip and usage, and interaction between pergore and usage into the model.
Before that, we first center the pergore and perAA variables.
Fit the new model with lm. We note the model respects the hierarchy. That is the main effects are automatically added to the model in presense of their interaction. Question: how to specify a formula involving just an interaction term but not their main effect?
How do we interpret \(\widehat \beta_{\text{cperAA}} = 0.0283\)?
How do we interpret \(\widehat \beta_{\text{equipOS-PC}} = 0.016\)?
How do we interpret \(\widehat \beta_{\text{usageurban}} = -0.019\)?
How do we interpret \(\widehat \beta_{\text{cpergore:usageurban}} = -0.009\)?
6 Hypothesis testing
We want to formally compare the two linear models.
A larger model \(\Omega\) with \(p=9\) parameters and
a smaller model \(\omega\) with \(q=3\) parameters.
The \(F\)-test compares the \(F\)-statistic \[
F = \frac{(\text{RSS}_{\omega} - \text{RSS}_{\Omega}) / (p - q)}{\text{RSS}_{\Omega} / (n - p)}
\] to its null distribution \(F_{p-q, n-p}\). The small p-value 0.0028 indicates we should reject the null model \(\omega\).
anova(lmod, lmodi)
Analysis of Variance Table
Model 1: undercount ~ pergore + perAA
Model 2: undercount ~ cperAA + cpergore * usage + equip
Res.Df RSS Df Sum of Sq F Pr(>F)
1 156 0.093249
2 150 0.081775 6 0.011474 3.5077 0.002823 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
We can carry out a similar \(F\)-test for each predictor in a model using the drop1 function. The nice thing is that the factors such as equip and cpergore * usage are droped as a group.
We also see \(F\)-test for quantitative variables, e.g., cperAA, conincides with the \(t\)-test reported by the lm function. Question: why drop1 function does not drop predictors cpergore and usage?
7 Confidence intervals
Confidence intervals for individual parameters can be construced based on their null distribution \[
\frac{\widehat{\beta}_j}{\text{se}(\widehat{\beta}_j)} \sim t_{n-p}.
\] That is a \((1-\alpha)\) confidence interval is \[
\widehat{\beta_j} \pm t_{n-p}^{(\alpha/2)} \text{se}(\widehat{\beta_j}).
\]
\(\mathbb{E}(\mathbf{Y}) = \mathbf{X} \boldsymbol{\beta}\), or equivalently, \(\mathbb{E}(\boldsymbol{\epsilon}) = \mathbf{0}\). That is we have included all the right variables and \(Y\) depends on them linearly.
Errors \(\epsilon_i\) are independent and normally distributed with common variance \(\sigma^2\). That is \(\widehat{\boldsymbol{\epsilon}} \sim \text{N}(\mathbf{0}, \sigma_0^2 \mathbf{I}_n)\).
We’d like to check these assumptions using graphical or numerical approaches.
Four commonly used diagnostic plots can be conveniently obtained by plot function.
plot(lmodi)
The residual-fitted value plot is useful for checking the linearity and constant variance assumptions.
The scale-location plot plots \(\sqrt{|\widehat{\epsilon}_i|}\) vs fitted values and serves similar purpose as the residual-fitted value plot.
The QQ plot checks for the normality assumption. It plots residuals vs the theoretical quantiles from a standard normal distribution \(\Phi^{-1}\left( \frac{i}{n+1} \right)\), \(i=1,\ldots,n\).
Residual-leverage plot. The fitted values are \[
\widehat{\mathbf{y}} = \mathbf{X} \widehat{\boldsymbol{\beta}} = \mathbf{X} (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{y} = \mathbf{H} \mathbf{y}.
\] The diagonal entries of the hat matrix, \(h_i = H_{ii}\), are called leverages. For example, \[
\text{Var}(\widehat{\boldsymbol{\epsilon}}) = \text{Var}(\mathbf{Y} - \mathbf{X} \widehat{\boldsymbol{\beta}}) = \text{Var} [(\mathbf{I} - \mathbf{H}) \mathbf{Y}] = (\mathbf{I} - \mathbf{H}) \text{Var}(\mathbf{Y}) (\mathbf{I} - \mathbf{H}) = \sigma^2 (\mathbf{I} - \mathbf{H}).
\] If \(h_i\) is large, then \(\text{var}(\widehat{\epsilon_i}) = \sigma^2 (1 - h_i)\) is small. The fit is “forced” to be close to \(y_i\). Points on the boundary of the predictor space have the most leverage.
The Cook distance is a popular influence diagnostic \[
D_i = \frac{(\widehat{y}_i - \widehat{y}_{(i)})^T(\widehat{y}_i - \widehat{y}_{(i)})}{p \widehat{\sigma}^2} = \frac{1}{p} r_i^2 \frac{h_i}{1 - h_i},
\] where \(r_i\) are the standardized residuals and \(\widehat{y}_{(i)}\) are the predicted values if the \(i\)-th observation is dropped from data. A large residual combined with a large leverage results in a larger Cook statistic. In this sense it is an influential point.
Let’s display counties with Cook distance \(>0.1\). These are those two counties with unusual large undercount.
Another useful plot to inspect potential outliers in positive values is the half-normal plot. Here we plot the sorted leverages \(h_i\) against the standard normal quantiles \(\Phi^{-1} \left(\frac{n+i}{2n + 1}\right)\). We do not expect a necessary straight line, just look for outliers, which is far away from the rest of the data.
# this function is available from faraway packagehalfnorm(hatvalues(lmodi), ylab ="Sorted leverages")
These two counties have unusually large leverages. They are actually the only counties that use paper ballot.
# A tibble: 2 × 16
county equip econ perAA usage atlanta gore bush other votes ballots
<chr> <fct> <fct> <dbl> <fct> <fct> <int> <int> <int> <int> <int>
1 MONTGOMERY PAPER poor 0.243 rural notAtlanta 1013 1465 31 2509 2573
2 TALIAFERRO PAPER poor 0.596 rural notAtlanta 556 271 5 832 881
undercount pergore perbush cpergore cperAA
<dbl> <dbl> <dbl> <dbl> <dbl>
1 0.0249 0.404 0.584 -0.00458 0.0000189
2 0.0556 0.668 0.326 0.260 0.353
9 Robust regression
In presence of outliers, if these outliers represent data entry errors, then we can simply remove these observations and proceed with linear regression.
If these outliers are real observations, then we can use robust linear regression.
Call: rlm(formula = undercount ~ cperAA + cpergore * usage + equip,
data = gavote)
Residuals:
Min 1Q Median 3Q Max
-6.026e-02 -1.165e-02 -6.587e-06 1.100e-02 1.379e-01
Coefficients:
Value Std. Error t value
(Intercept) 0.0414 0.0023 17.8662
cperAA 0.0327 0.0254 1.2897
cpergore -0.0082 0.0418 -0.1972
usageurban -0.0167 0.0038 -4.4063
equipOS-CC 0.0069 0.0038 1.8019
equipOS-PC 0.0081 0.0048 1.6949
equipPAPER -0.0059 0.0138 -0.4269
equipPUNCH 0.0170 0.0055 3.0720
cpergore:usageurban 0.0073 0.0316 0.2298
Residual standard error: 0.01722 on 150 degrees of freedom
Notably the regression coefficient for equipOS-PC changed from 0.0156 to 0.0081. It downweights the influence of two observations with equi being OS-PC.
p-values are not reported because inference for robust model is much harder than regular linear regression.
10 Transformation
Transformation of variables can alleviate violation of certain assumptions.
In this case, transformation of responseundercount is tricky because
minimum value of undercount is 0, precluding certain transformations such as log and inverse, and
after transformation the interpretation of regression coefficients becomes hard.
Transformations of predictors are less problematic.
Consider using orthogonal polynomials on the predictor cperAA.
`termplot` graphically summarizes the effect of each predictor. `terms` argument of `termplot` function specifies which term to plot. Pratial residuals are predictor values plus residual values.
Variable selection concerns the problem of selecting the best set of variables to put in a model. There are several approaches and it is still an active research area.
Exhaustive search, also called all-subset regression, enumerates all \(2^p\) submodels and selects the best one according to a criterion such as the adjusted \(R^2\). The regsubsets function in the leaps package implements this search. The implementation only applies to quantitative predictors.
When there are many predictors, all-subset regression quickly becomes computationally infeasible. Stepwise regression does heuristic search. A popular criterion is the Akaike Information Criterion (AIC) \[
\text{AIC} = - 2 \cdot \text{maximum log-likelihood} + 2p,
\] where \(p\) is the number of parameters. We start with a big model that includes all two-way interactions between qualitative predictors and all two-way interactions between qualitative and quantitative predictors
Call:
lm(formula = undercount ~ equip + econ + perAA + equip:econ +
equip:perAA, data = gavote)
Residuals:
Min 1Q Median 3Q Max
-0.057011 -0.010632 -0.000069 0.009198 0.082545
Coefficients: (2 not defined because of singularities)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.041871 0.005028 8.328 6.5e-14 ***
equipOS-CC -0.011327 0.007373 -1.536 0.126700
equipOS-PC 0.008575 0.011178 0.767 0.444288
equipPAPER -0.058427 0.037014 -1.579 0.116687
equipPUNCH -0.015751 0.018745 -0.840 0.402175
econpoor 0.020266 0.005529 3.665 0.000349 ***
econrich -0.016966 0.012392 -1.369 0.173128
perAA -0.042040 0.016594 -2.534 0.012385 *
equipOS-CC:econpoor -0.010964 0.009885 -1.109 0.269224
equipOS-PC:econpoor 0.048385 0.013795 3.507 0.000608 ***
equipPAPER:econpoor NA NA NA NA
equipPUNCH:econpoor -0.003560 0.012427 -0.286 0.774921
equipOS-CC:econrich 0.002278 0.015378 0.148 0.882465
equipOS-PC:econrich -0.013318 0.017054 -0.781 0.436149
equipPAPER:econrich NA NA NA NA
equipPUNCH:econrich 0.020031 0.021997 0.911 0.364045
equipOS-CC:perAA 0.107249 0.032855 3.264 0.001377 **
equipOS-PC:perAA -0.005906 0.043414 -0.136 0.891981
equipPAPER:perAA 0.129136 0.081806 1.579 0.116676
equipPUNCH:perAA 0.086849 0.046500 1.868 0.063875 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.02 on 141 degrees of freedom
Multiple R-squared: 0.4276, Adjusted R-squared: 0.3585
F-statistic: 6.195 on 17 and 141 DF, p-value: 1.318e-10
The coefficient for equipPAPER:econrich is not estimated because there are no counties that are rich and use paper ballot. The corresponding column in \(\mathbf{X}\) matrix is all zeros.
The coefficient for equipPAPER:econpoor is not estimated because there are only two counties that use paper ballot and both of them are poor. So the corresponding colmuns for equipPAPER and equipPAPER:econpoor are identical. In other words these two predictors are not identifiable. Therefore only equipPAPER is estimated but not equipPAPER:econpoor.
gavote %>%filter(equip =="PAPER")
# A tibble: 2 × 16
county equip econ perAA usage atlanta gore bush other votes ballots
<chr> <fct> <fct> <dbl> <fct> <fct> <int> <int> <int> <int> <int>
1 MONTGOMERY PAPER poor 0.243 rural notAtlanta 1013 1465 31 2509 2573
2 TALIAFERRO PAPER poor 0.596 rural notAtlanta 556 271 5 832 881
# ℹ 5 more variables: undercount <dbl>, pergore <dbl>, perbush <dbl>,
# cpergore <dbl>, cperAA <dbl>
Let’s attempt predictions at all combination of levels of econ and equip at median perAA
equip
propAA LEVER OS-CC OS-PC PAPER PUNCH
low 0.037 0.038 0.045 -0.007 0.031
medium 0.032 0.046 0.039 0.003 0.036
high 0.027 0.053 0.034 0.014 0.042
13 Conclusion
BACKGROUND: The controversy surrounding the 2000 US presidential election highlighted the potential impact of voting equipment on voting errors. However, most existing studies have focused on examining voting equipment across states, rather than within individual states, potentially leading to aggregation bias. Furthermore, social-economic status and race have not been adequately considered in previous research.
OBJECTIVE: This study aimed to identify the factors influencing voting errors in the state of Georgia, taking into account social-economic status and race.
METHOD: Data from the Georgia Secretary of State’s Election Administration Web site for the 2000 general election were used to calculate the percentage of unrecorded votes over the total ballots issued in each county. A linear regression analysis was performed to explore the factors contributing to undercounts.
RESULTS: The study found that, for the 2000 general election in Georgia, the percentage of African American residents and the economic status of the county were significantly associated with undercount, while the type of voting equipment was not a significant predictor. Specifically, using the OS-PC voting equipment in counties with poor economic status was associated with 0.04 (95% CI: [insert confidence interval here]) higher undercount compared to counties with medium economic status. Additionally, for counties using OC-CC compared to LEVER equipment, an increase in the percentage of African American population was associated with a statistically significant higher undercount.
CONCLUSIONS: In conclusion, this study reveals that social-economic status and race were the major contributors to voting errors in Georgia, while voting equipment type did not emerge as a significant predictor overall. However, our findings suggest that the use of certain types of voting equipment, such as the OS-PC and OS-CC, may lead to higher voter errors in counties with poor economic status and higher African American populations. Overall, these results underscore the importance of considering a range of factors when investigating voting errors in the US election system.
